Trading Monotonicity Demands versus Eciency

The present paper deals with the learnability of indexed families L of uniformly recursive languages from positive data. We consider the in uence of three monotonicity demands and their dual counterparts to the e ciency of the learning process. The e ciency of learning is measured in dependence on the number of mind changes a learning algorithm is allowed to perform. The three notions of (dual) monotonicity re ect different formalizations of the requirement that the learner has to produce better and better (specializations) generalizations when fed more and more data on the target concept. We distinguish between exact learnability (L has to be inferred with respect to L), class preserving learning (L has to be inferred with respect to some suitably chosen enumeration of all the languages from L), and class comprising inference (L has to be learned with respect to some suitably chosen enumeration of uniformly recursive languages containing at least all the languages from L). In particular, we prove that a relaxation of the relevant (dual) monotonicity requirement may result in an arbitrarily large speed-up. However, whether or not such a speed-up may be achieved crucially depends on the set of allowed hypothesis spaces as well as of the (dual) monotonicity demands involved.